variable groupoid

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curve

Definition 0.1.

A variable groupoid is defined as a family of groupoids
{𝖦λ}indexed by a parameter λ∈T , with T being either an index set or a class (which may be a time parameter, for time-dependent or dynamic groupoids). If λ belongs to a set M, then we may consider simply a projection𝖦×M⟶M, which is an
example of a trivialfibration. More generally, one can consider a fibration of groupoids𝖦↪Z⟶M (Higgins and Mackenzie, 1990) as defining a non-trivial variable groupoid.

Remarks
An indexed family or class of topological groupoids[𝖦i] with i∈I in the categoryGrpd of groupoids
with additional axioms, rules, or properties of the underlying topological groupoids,
that specify an indexed family of topological groupoid homomorphisms for each variable groupoid
structure.

Besides systems modelled in terms of a fibration of groupoids,
one may consider a multiple groupoid defined as a set of N
groupoid structures, any distinct pair of which satisfy an
interchange law which can be formulated as follows.
There exists a unique expression with the following content:

where i and j must be distinct for this concept to be well defined.
This uniqueness can also be represented by the equation

(x∘jy)∘i(z∘jw)=(x∘iz)∘j(y∘iw).

(0.2)

Remarks
This illustrates the principle that a 2-dimensional formula may be
more comprehensible than a linear one.

Brown and Higgins, 1981a, showed that certain multiple groupoids
equipped with an extra structure called connections were
equivalent to another structure called a crossed complex
which had already occurred in homotopy theory. such as
double, or multiple groupoids (Brown, 2004; 2005).
For example, the notion of an atlas of structures should,
in principle, apply to a lot of interesting, topological and/or
algebraic, structures: groupoids, multiple groupoids, Heyting
algebras, n-valued logic algebras and C*-convolution
-algebras. Such examples occur frequently inHigher Dimensional Algebra
(HDA).